Cutting Corners

Domains

The two psychedelic designs up above arise from their simplistic ancestors we looked at last time by cutting off corners. These are still two conformal annuli that also satisfy a slightly complicated condition on the lengths of their edges, which makes them responsible for a variation of the Diamond surface:

 

 

Mathematica

If you cut either of the psychedelic shapes into quarters, using a vertical and a horizontal cut, you get four right angled octagons, with some right angles being exterior angles. Similarly, the marked symmetry lines on the surface up above cut the surface into eight right angled curved octagons, that correspond to the psychedelic octagons via a conformal and harmonic map.  

 

D5 deg1

There is a 1-parameter family of such critters. Above and below are larger portions of extreme cases that also show how the surface repeats.

D5 deg2

You can see in the image above pieces of the doubly periodic Karcher-Scherk surface reappearing. No surprise, its psychedelic polygons also arise by cutting corners in the polygons corresponding to the original Scherk surface.

Everything simple reappears.

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This Is Not a Helicoid

But almost. It has a vertical axis, lots of horizontal lines, and it twists.

Nothelicoid

But it is part of something bigger, a triply periodic minimal surface. 32 copies of the above piece, replicated by rotations and reflections, look like this:

Nothelicoidcopies

This surface sits in a rectangular box over a square. If you identify top and bottom edge of the original squarical helicoid, you get a doubly twisted annulus, which is intimately (confomally, that is) related to a hollow spiderweb:

D spider 01

 

 

When squeezing the height down, our non-helicoids become even more helicoidal. When pulling the height up, the helicoids disappear. What we have here is a deformation of the Diamond surface of Hermann Amandus Schwarz.

When he sees this, he will probably just nod.

One

What happens when we pull a little further? We see doubly periodic Scherk surfaces emerging, stacked on top of each other.

Triplyscherk

 

 

Golem (Fattend Skeletons)

Today I want to look at decorations of simplicial graphs. As an example, here is a decoration of last week‘s skeleton:

CI3

The vertices of the two skeletons have been replaced by tetrahedra, oriented and scaled so that vertices of neighbors touch. This should explain what I mean by a decoration: A geometric construction that consistently modifies the graph, in order to obtain something with similar symmetries (many) and possibly other desired properties. Another simple and well known decoration is that by Voronoi cells: We replace each vertex by the set of all points that are closer to that vertex than to any other vertex. In this case, the Voronoi cells are truncated octahedra, as shown in another post. Instead of replicating it here, one can also pass to the dual tiling, which is by rhombic dodecahedra. Try viewing it cross eyed.

StereoD

This is another polyhedral version of the Diamond surface of Schwarz. Like the one obtained from the truncated dodecahedra, the polyhedral approximation shares the conformal structure (by sheer symmetry).

There is another decoration that is quite remarkable:

CI4

Here, we have replaced each vertex and each edge of the original graph by an octahedron, properly scaled an oriented. We thus get two fattened skeletons that are congruent and disjoint. All faces are equilateral triangles, and all vertices have valency 6. There even is enough room between them to fit in the truncated octahedra, as one can see in the last image.

CI5

This image also shows how to position each octahedron within the cubical lattice: The central octahedron has its vertices along the edges of the lattice, dividing each edge in the ratio 1:3. That ratio ensures that the other three octahedra in the image become in fact regular. More about this next week.

Walls and Connections

The cubical lattice is a seemingly simple way to arrange spheres in space. By connecting spheres that are closest to each other, we get a line configuration I have also written about before.

c

Let’s increase the complexity by adding another copy of the same configuration, shifted by 1/2 of a unit step in all coordinate directions. This is sometimes called the body-centered cubical Bravais lattice.

CI

We can also recognize here the two skeletal graphs of the two components of the complement of the Schwarz P minimal surface. This means that the P surface will separate the yellow and the red lattices.

Now we would like to connect the two separate systems of spheres with each other. Note that each yellow sphere is surrounded by 8 red spheres (and vice vera), at the vertices of a cube centered at the yellow sphere. This suggests to connect the yellow center to just four of these red neighbors, by choosing the vertices of a tetrahedron, as to obtain a 4-valent graph. Like so:

CI2

While this is still simple, it starts to look confusing. The new skeleton has again two components, and again they can be separated by a classical minimal surface, the Diamond surface of Hermann Amandus Schwarz.

All this should remind us of the Laves graphs, which are skeletal graphs of the gyroid.

Tetra D

You can see that these skeletal graphs have girth 6. Below is a larger piece of the D-surface. Everything here is triply periodic and very symmetric. In contrast to the Laves graph, these here have no chirality.

D skeleton

Next week, we will decorate these skeletons a little.